Fundamental solution.

A fundamental solution for the partial derivatives operator
is the distribution
with the following
property

Note that the equation

has the
solution
where the * is the convolution
operation

Indeed, since differentiation commutes with
convolution,

Example

(Fundamental solution for ODE).
Consider the ODE
operator
where the
and
are functions
.
We conjecture a fundamental solution of the
form
where the
is the step function (see the formula (
step
function
)) and the
is some unknown smooth function. We substitute
into
and calculate the
derivatives
The functions
are supported at 0. Hence, if we
set
then
We conclude that the
is defined by the following Cauchy
problem